High order numerical simulation of non-Fourier heat conduction: An application of numerical Laplace transform inversion ☆ Iman Rahbari, Farzam Mortazavi, Mohammad Hassan Rahimian ⁎ School of Mechanical Engineering, University College of Engineering, University of Tehran, Tehran, Iran abstract article info Available online 20 December 2013 Keywords: Non-Fourier conduction Insulated boundaries Finite slab Numerical Laplace transform inversion Laplace solution Dirac function Step function Non-Fourier heat conduction phenomenon in a finite slab with insulated boundaries is investigated in the present paper. Since solving the hyperbolic heat conduction equation analytically requires considerable effort, a new high-order numerical approach has been implemented to achieve comparable exactitude. This method solves the considered equation in Laplace space and numerical inversion is employed with the intention of trans- formation to temporal domain. In order to examine numerical accuracy of this method, Dirac delta heat flux is applied to the assumed medium and results were compared with those of the analytical solution. It was observed that numerical values follow exact ones, at least up to the seventh order of accuracy. In addition, Step and Trian- gular heat pulses in the medium were studied to reveal temporal and spatial non-Fourier heat conduction char- acteristics. It was found that in large values of Ve number, for various kinds of heat fluxes carrying the same amount of energy, temperature distribution varies conspicuously through the medium; nevertheless, at each pass of heat wave, a specific point experiences a definite rise of temperature regardless of the type of heat flux provided that the same conditions are present. © 2013 Elsevier Ltd. All rights reserved. 1. Introduction Fourier heat conduction model assumes an infinite propagation speed for thermal disturbances, i.e. when a temperature gradient applies to a medium, everywhere feel it immediately. But in reality, the maximum speed for transportation of a phenomenon is limited to the light speed. Thus, it is clear that disturbances in conduction mode of heat transfer propagate in a finite speed. However Fourier's assumption may work for many industrial appli- cations; deviation from this model is considerable in many areas in which laser radiation either in medical or industrial purposes, analysis of solar collector plates, applications subjected to high heat fluxes, and heat conduction in the very low ambient temperature are among them [1–4]. Many researches have been carried out to achieve an appropriate non-Fourier heat conduction model which makes up this deviancy and captures experimental evidences in the mentioned areas. The most frequent model was proposed by Cattaneo [5] and Verenotte [6], which simply takes into account the finite speed of heat propagation by means of a first order Taylor series expansion of flux vector in time and consequently adding a lag term in flux equation, as follow: q þ τ ∂q ∂t ¼ -k∇T ð1Þ where q is flux vector and τ is thermal relaxation time which depends on the characteristics of employed material. Combination of Eq. (1) and energy equation yields: ∂T ∂t þ τ ∂ 2 T ∂t 2 ¼ α∇ 2 T ð2Þ where α is the thermal diffusivity and C h ¼ ffiffi α τ p is the speed of heat wave propagation. In this way, if τ → 0, Fourier heat conduction equation will be recovered. It is obvious that Eq. (2) is hyperbolic in nature unlike the classical heat conduction which is parabolic. Several experimental researches have been conducted to study governing equation of non-Fourier heat conduction and among them, the study of Jackson and Walker [7] on NaF thermal conductivity, sec- ond sound and Phonon–Phonon interaction at very low temperature, the observation of second sound in Bismuth by Narayanamurti and Dynes [8], the work of Roetzel et al. [9] on the materials with non- homogeneous inner structure, and more recently the research of HaiDong et al. [10] on the heat conduction in metallic nanofilms from large currents at low temperatures could be mentioned here. In the other side, numerous studies have been done using analytical and numerical approaches from the earliest ages of introducing governing equation of non-Fourier heat conduction. Considerable quan- tities of them deal with one-dimensional form of this equation. Tang and Araki [11] studied non-Fourier heat conduction in a finite slab with one isolated boundary and the other subjected to a periodic heat flux by implying Laplace transform analytically. A similar problem is considered by Abdel-Hamid [12] and solved by means of Integral International Communications in Heat and Mass Transfer 51 (2014) 51–58 ☆ Communicated by W.J. Minkowycz ⁎ Corresponding author. E-mail address: rahimyan@ut.ac.ir (M.H. Rahimian). 0735-1933/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.icheatmasstransfer.2013.12.003 Contents lists available at ScienceDirect International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt